Getting Biblical in Daily Life
Jewish Perspective
Since Proverbs was originally compiled by Israelites—though some of the actual sayings clearly come from non-Israelite sources (Egyptian, etc.)—it's clearly been pretty important to Jewish tradition.
Not only has it influenced apocryphal works like Ecclesiasticus and The Wisdom of Solomon, but also the "Pirke Avoth"—the "Sayings of the Fathers"—which records the words of great Rabbis roughly contemporary with Jesus. You can even see the vague ghost of its influence creeping into modern works like the aphorisms of Franz Kafka.
The wisdom of Proverbs, stressing hard work, steadfastness, honesty, and concern for the poor is reflected in these later Jewish traditions. In the "Pirke Avoth," the great Rabbi Tarphon says, "You do not need to complete the work, but neither are you free to desist from it." It also includes such famous sayings as Rabbi Hillel's short and pithy, "If not now, when?"
Christian Perspective
In the early days of the Christian Church, there was a huge debate over who exactly Jesus was: was he God, or was he a sort of angelic being created by God? A significant part of this debate was closely related to the idea that Christ existed before the creation of the world—like in the beginning of the Gospel of John. This makes Jesus pretty similar to Wisdom, who also (according to Proverbs) existed before the world was made.
The people who believed that Jesus wasn't God, but was actually the best created being of all, were called the "Arians." They lost out in this debate, in the end—partially because of reasons related to Proverbs.
Jesus as the "Word" of God was seen as being closely connected to, or the same as, the "Wisdom" of God from Proverbs (even though Proverbs' Wisdom has a much more feminine vibe). The opponents of the Arians won because they argued that Jesus wasn't created like everything else, but was rather "begotten" directly by God. (Psst, if you don't understand what this means or what the difference is, you're in good company—theologians have been debating these questions for centuries).
Islamic Perspective
On the one hand, Islam seems to have been minimally influenced by The Book of Proverbs (or, at least, not that many scholars are writing about how it was influenced), but it seems to have been strongly influenced by the Near Eastern and Jewish wisdom tradition that Proverbs represented, as a whole.
While the Qur'an provides plenty of revelations, moral admonitions, and legal codes, the sayings of the Prophet Muhammad—or Hadith—contain a practical wisdom tradition of their own, similar in many ways to that of Proverbs. The Hadith offers up plenty of down-to-earth advice that would be well in place within Proverbs' pages: "Trust in God but tie your camel's leg," for instance.